Page 259 - Electromagnetics

assume that as S →∞ there are no sources external to S and thus no need for the
boundary fields. This is indeed the case. If all sources are localized, the fields they
produce behave in just the right manner for the surface integral in (4.165) to vanish, and
thus uniqueness is again guaranteed. Later we will find that the electric and magnetic
fields produced by a localized source at great distance have the form of a spherical wave:
e − jkr
ˇ ˇ
E ∼ H ∼ .
r
If space is taken to be slightly lossy, then k is complex with negative imaginary part, and
thus the fields decrease exponentially with distance from the source. As we argued above,
it may not be physically meaningful to assume that space is lossy. Sommerfeld postulated
that even for lossless space the surface integral in (4.165) vanishes as S →∞. This has
been verified experimentally, and provides the following restrictions on the free-space
fields known as the Sommerfeld radiation condition:
ˇ
ˇ

lim r η 0 ˆ r × H(r) + E(r) = 0, (4.167)
r→∞
ˇ
ˇ

lim r ˆ r × E(r) − η 0 H(r) = 0, (4.168)
r→∞
where η 0 = (µ 0 / 0 ) 1/2 . Later we shall see how these expressions arise from the integral
solutions to Maxwell’s equations.
4.10.2 Reciprocity revisited
In § 2.9.3 we discussed the basic concept of reciprocity, but were unable to examine
its real potential since we had not yet developed the theory of time-harmonic fields. In
this section we shall apply the reciprocity concept to time-harmonic sources and fields,
and investigate the properties a material must display to be reciprocal.

The general form of the reciprocity theorem. As in § 2.9.3, we consider a closed
surface S enclosing a volume V . Sources of an electromagnetic field are located either
inside or outside S. Material media may lie within S, and their properties are described
in terms of the constitutive relations. To obtain the time-harmonic (phasor) form of the
reciprocity theorem we proceed as in § 2.9.3 but begin with the phasor forms of Maxwell’s
equations. We find
ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ
∇· (E a × H b − E b × H a ) = j ˇω[H a · B b − H b · B a ] − j ˇω[E a · D b − E b · D a ] +
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
+ [E b · J a − E a · J b − H b · J ma + H a · J mb ], (4.169)
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
where (E a , D a , B a , H a ) are the fields produced by the phasor sources (J a , J ma ) and (E b , D b , B b , H b )
ˇ
ˇ
are the fields produced by an independent set of sources (J b , J mb ).
As in § 2.9.3, we are interested in the case in which the first two terms on the right-
hand side of (4.169) are zero. To see the conditions under which this might occur, we
substitute the constitutive equations for a bianisotropic medium
ˇ
ˇ
ˇ
˜ ¯
D = ξ · H + ˜ ¯ · E,
˜ ¯
ˇ
ˇ
ˇ
B = ˜ ¯µ · H + ζ · E,
into (4.169), where each of the constitutive parameters is evaluated at ˇω. Setting the
two terms to zero gives

ˇ ˇ ˜ ¯ ˇ ˇ ˇ ˜ ¯ ˇ
j ˇω H a · ˜ ¯µ · H b + ζ · E b − H b · ˜ ¯µ · H a + ζ · E a −
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